We systematically explore and show the existence of finite-temperature continuous quantum phase transition (CTQPT) at a critical point, namely, during solidification or melting such that the first-order thermal phase transition is a special case within CTQPT. Infact, CTQPT is related to chemical reaction where quantum fluctuation (due to wavefunction transformation) is caused by thermal energy and it can occur maximally for temperatures much higher than zero Kelvin. To extract the quantity related to CTQPT, we use the ionization energy theory and the energy-level spacing renormalization group method to derive the energy-level spacing entropy, renormalized Bose-Einstein distribution and the time-dependent specific heat capacity. This work unambiguously shows that the quantum phase transition applies for any finite temperatures.
We systematically explore and show the existence of finite-temperature continuous quantum phase transition (CTQPT) at a critical point, namely, during solidification or melting such that the first-order thermal phase transition is a special case within CTQPT. Infact, CTQPT is related to chemical reaction where quantum fluctuation (due to wavefunction transformation) is caused by thermal energy and it can occur maximally for temperatures much higher than zero Kelvin. To extract the quantity related to CTQPT, we use the ionization energy theory and the energy-level spacing renormalization group method to derive the energy-level spacing entropy, renormalized Bose-Einstein distribution and the time-dependent specific heat capacity. This work unambiguously shows that the quantum phase transition applies for any finite temperatures.
The dynamical critical exponent is introduced. The phase diagram and the phase transitions in the Bose–Hubbard model of interacting bosons on a lattice are determined. The concept of quantum fluctuations is introduced on the example of an interacting superfluid, and finally the special scaling of conductivity is discussed. Dynamical critical exponent The finite temperature phase transitions studied in previous chapters are the result of the competition between the entropy and the energy terms in the free energy: the weight of entropy increases with temperature, and ultimately destroys the order that may be existing in the system. A sharp phase transition between two phases exhibiting qualitatively different correlations may, however, occur even at zero temperature, by varying a coupling constant in the Hamiltonian. The transition then corresponds to a non-analyticity of the energy of the ground state. A simple example is provided by the interacting bosons, where the superfluid transition may be brought about by tuning the chemical potential at T = 0. Such T = 0 phase transitions are called quantum phase transitions , and will be the subject of the present chapter. In general, a quantum phase transition may lie at the end of a line of thermal phase transitions, as in the bosonic example mentioned above. It is possible, however, that the system may not even have an ordered state at finite temperatures, but still exhibits a quantum critical point. Two different situations are depicted in Fig. 8.1. Examples of such phase diagrams are provided by the system of interacting bosons which will be discussed in Sections 8.2 and 8.3.
Aim and Scope of this Book A plethora of systems exhibit phase transitions as the temperature or some other parameter is changed. Examples range from the ice-water phase transition observed in our daily life to the loss of ferromagnetism in iron or to the more sophisticated Mott insulator-superfluid phase transition observed in optical lattices [343]. The last five decades have witnessed a tremendous upsurge in the studies of phase transitions at finite temperature [727, 149, 333, 136, 494, 541, 556]. The success of Landau-Ginzburg theories and the concepts of spontaneous symmetry breaking and the renormalization group [27, 410, 821, 578] in explaining many of the finite temperature phase transitions occurring in nature has been spectacular. In this book, we will consider only a subclass of phase transitions called quantum phase transitions (QPTs) [154, 658, 725, 799, 185, 63, 62, 66, 141, 744] and we will discuss these mainly from the view point of recent studies of information and dynamics. QPTs are zero temperature phase transitions which are driven by quantum fluctuations and are usually associated with a non-analyticity in the ground state energy density of a quantum many-body Hamiltonian. We will focus on continuous QPTs where the order parameter vanishes continuously at the quantum critical point (QCP) at some value of the parameters which characterize the Hamiltonian. We will not discuss first order quantum phase transitions associated with an abrupt change in the order parameter. Usually, a first order phase transition is characterized by a finite discontinuity in the first derivative of the ground state energy density. A continuous QPT is similarly characterized by a finite discontinuity, or divergence, in the second derivative of the ground state energy density, assuming that the first derivative is continuous. This is of course the classical definition; we will later mention some QPTs where the ground state energy density is not necessarily singular.
Discussing quantum phase transition and quantum critical point Main Viewpoints and Conclusions: In physics, a quantum phase transition (QPT) is a phase transition between different quantum phases (phases of matter at zero temperature). Contrary to classical phase transitions, quantum phase transitions can only be accessed by varying a physical parameter—such as magnetic field or pressure—at absolute zero temperature. The transition describes an abrupt change in the ground state of a many-body system due to its quantum fluctuations.[1] In fact and details, the study of the quantum phase transition and quantum critical point, is the study about of a material in a condition which with a definite electromagnetic and pressure environment, and when the temperature continues change to lower, until reaches absolute zero temperature, the properties changing of this material with the changes of the environment temperature (whether it is a single kind of atom or more kinds of atoms of the crystal compound, and especially with regards to solid material). With a definite electromagnetic and pressure environment, the environment temperature continues change to lower, until reaches absolute zero temperature, in the process, the material lost its energy continuously, so, the energy both of the all extranuclear charges and all nucleuses (main is π-mesons body) of the material also is go into further lower continuously. Meanwhile, the volume density, natural frequency and energy level of the extranuclear charges go into further lower continuously; the difference of the volume charge density, natural frequency and energy level between all parts of the material becomes to smaller, and reaches the same and consistency at absolute zero temperature. In the state, all atoms of the material at the same and lowest energy level of their ground state, the extranuclear-charges of all atoms with the same density and polymerized into a single complete whole and agglomerate together with all nucleuses, namely, the material is being its Bose-Einstein condensate (BEC).[2][3][4][5] The process of the volume density, natural frequency and energy level of the extranuclear charges go into further lower continuously; the difference of the volume charge density, natural frequency and energy level between all parts of the material becomes smaller, also is a process that the energy levels of every parts of the extranuclear charges goes into more lower state which higher than the lowest energy level-ground state, and this is the weak first-order quantum phase transition which we called that the energy levels from higher to lower but still higher than the lowest energy level-ground state yet; further, we called the quantum state transition is second-order quantum phase transition which the energy levels of every parts of the extranuclear charges from many different higher energy levels goes into the same and lowest energy level-ground state after the material reaches into and at its Bose-Einstein condensate; and the environment condition (combined effect generated by magnetic-field, pressure and temperature) that is the quantum phase dividing line or called quantum critical point which makes the material reaches into its Bose-Einstein condensate and the all extranuclear charges from the state with many different volume charge densities, natural frequencies and energy levels transform into a single complete whole body which with the same and unique volume charge density, natural frequency and energy level at its ground state; or, in other words, the quantum critical point is the dividing line of the two state that the all extranuclear charges with many different volume charge densities, natural frequencies and energy levels or the all extranuclear charges with the same and unique volume charge density, natural frequency and energy level at its ground state. Contrary to the case of a material into its Bose-Einstein condensate, after temperature rise and beyond the quantum critical point, the material would deviate from its Bose-Einstein condensate, the extranuclear charge-body separates from the all nucleuses, and the all nucleuses returns to their free states. Same time, the volume density of every parts of the extranuclear charge-body whether combined into atomic bonds or not, all changes into and at non same and unique state, the energy levels of the extranuclear charge-body appear as separate and in a discrete state, and then restoration and presentation quantum behaviors in the microscopic scale of the material. References & Related stories [1] Quantum phase transition https://en.wikipedia.org/wiki/Quantum_phase_transition [2] Bose-Einstein condensate https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_condensate [3] A New Model of Atomic Structure http://vixra.org/abs/1401.0147 [4] The Causes and Mechanism of Atomic Energy Levels quantum http://vixra.org/abs/1402.0104 [5] The Structure, State and Properties of Matter in Bose-Einstein Condensate http://vixra.org/abs/1509.0184 [6] Critical behavior at a dynamic vortex insulator-to-metal transition Science, 2015; 349 (6253): 1202 Doi: 10.1126/science.1260507 [7] Inhomogeneity of charge-density-wave order and quenched disorder in a high-Tc superconductor Nature, 525, 359–362 (17 September 2015) Doi: 10.1038/nature14987 [8] The mechanism of charge density wave in Pt-based layered superconductors: SrPt2As2 and LaPt2Si2 Nature, scientific Reports 5, 15052(2015) Doi: 10.1038/srep15052 YuQuan Road, Zhao Feng Yuan Section, 3rd block, Feng Tai District, Beijing, CHINA
Quantum Phase Transitions Quantum phase transitions, in contrast to temperature-driven critical phenomena, occur due to a competition between different parameters describing the basic interactions of the system. Their specific feature is the quantum character of the critical fluctuations. This implies, through the uncertainty principle, that energy fluctuations and time are coupled. Then at zero temperature time plays a crucial and fundamental role, the static properties being entangled with the dynamics (Continentino, 1994a; Sondhi et al ., 1997; Sachdev, 1999). In this book we are mainly interested in quantum phase transitions that occur in electronic many-body systems and how scaling concepts can be useful to understand their properties close to these transitions (Continentino, 1994a). Even though a similar approach has been used in the case of interacting bosons (Fisher et al ., 1989) the fermionic problem has its own idiosyncrasies and difficulties. For example, there is no natural order parameter associated with the localisation transition in the electronic case, while bosons at zero temperature are either localised or superfluid so that the superfluid order parameter can be used to distinguish between both phases. If the results of the study of quantum phase transitions were restricted to zero temperature, this would be an interesting but purely academic area of research. What is really exciting about this subject is the effect of quantum critical points (QCP) in the finite temperature phase diagram of actual physical systems, even far away from the QCP (Freitas, 2015). As we will show, there is a special line in this phase diagram, the quantum critical trajectory , where the temperature dependence of the thermodynamic and transport properties is governed by the quantum critical exponents, i.e. those associated with the QCP. This and the observed crossover effects induced by temperature in the non-critical side of the phase diagram of a material with a QCP are sufficient to make the study of quantum phase transitions an inevitable subject. We start this chapter by introducing the scaling theory of quantum critical phenomena, since scaling concepts are used throughout this book.
Establishing the nature of a quantum phase transition in finite-size simulations -- whether continuous, first-order, or weak first-order -- is a fundamental challenge in quantum many-body computation. Especially, the weak first-order phase transition is affected by a super large correlation length and always displays as a continuous critical point in simulated finite-sizes. The core difficulty lies in the fact that there is no effective finite-size theory to distinguish these phase transitions in the realistic simulations limited by the computational resource. In this work, we have fixed this problem by introducing a unified finite-size framework based on the Renyi thermal entropy (RTE) and its derivative (DRTE) to detect and characterize quantum phase transitions. We derive complete scaling theories for the RTE and DRTE at second-order, first-order, and weak first-order transitions, showing that the DRTE naturally isolates the singular part of the free energy and strengthens the characteristics of various phase transitions in finite sizes. Using quantum Monte Carlo simulations, we demonstrate accurate data collapse and extraction of critical exponents at (2+1)-dimensional O(N) critical points. More importantly, the DRTE provides a smoking-gun signature of weak first-order transitions through a clear double-peak structure and a crossing at zero, which we unambiguously observe in debated deconfined quantum criticality candidates such as the J--Q models. Our approach offers a general, unbiased, and numerically efficient tool for probing the universal properties of quantum phase transitions, resolving long-standing ambiguities between continuous and weak first-order scenarios.
Introduction The quantum phase transitions considered so far in this book are mostly continuous, second order. They are associated with the divergence of a correlation length and of a characteristic time that renders the quantum critical point scale invariant. However, there are many important phase transitions at which the order parameter goes abruptly to zero and, more important, at which the characteristics of length and critical time do not diverge. In general there is also a latent heat , or work required, to transform one phase into another. In this and the next chapter we study these discontinuous quantum phase transitions. We proceed by considering initially some important cases of what is known as fluctuation induced quantum phase transitions to next treat a more general case of a first-order quantum phase transition. We will show through the study of these problems that the concepts of critical exponents and scaling remain useful, even though these systems do not exhibit scale invariance at their critical point. In Chapter 15 we present the scaling theory of first-order quantum phase transitions and use its results to treat some general problems. One of the most interesting cases of fluctuation-induced transitions occurs in superconductors due to the coupling of the charged carriers to the electromagnetic field. We consider initially a wider class of systems with macroscopic wave functions, the superfluids, of which superconductors are the charged version. We adopt a macroscopic approach to this problem and describe the superfluid by a complex order parameter φ , such that | φ | 2 is proportional to the density of condensed particles. The zero temperature phase diagram is expressed in terms of a control parameter, m 2 , such that, at m 2 = 0, the system presents a quantum phase transition from a superfluid state, for m 2 < 0, φ ≠ 0, to a normal state for m 2 > 0 where φ = 0. In the case of the superconductor with particles of charge q = 2 e , we will include the coupling of these particles to the electromagnetic field and study its effect in the phase diagram of these systems. We show that in the neutral superfluid, the zero temperature transition to the normal state is continuous and occurs at a quantum critical point.